$V$ and $W$ be two vector spaces such that $\dim V > \dim W$, then $T:V\longrightarrow W$ can't be Injective.

Question

"Let $V$ and $W$ be two vector spaces such that $\dim V > \dim W$ and let $T:V\longrightarrow W$ be a function between $V$ and $W$. If this is the case then $T$ can't be injective".

I know this statement is true if T is a linear map, but is it also true for any function $T$, linear or not?

surjective

@JoséCarlosSantos sorry I made a mistake while writing the statement. I already edited it — Eduardo Magalhães, May 01 '20 at 22:09
No, it doesn't have have to be surjective. What about $T\colon\Bbb R^2\longrightarrow\Bbb R$ defined by $T(x,y)=0$? — José Carlos Santos, May 01 '20 at 22:09
@JoséCarlosSantos I'm so sorry... I mistook the concepts of injectivity and Surjectivity... — Eduardo Magalhães, May 01 '20 at 22:14

Martin Argerami · Accepted Answer · 2020-05-01T22:19:11.590

3

(answer to version 2 of the question)

As stated, it is not true, even for linear maps. For instance take $V=\mathbb R^2$, $W=\mathbb R$, and $T(x,y)=0$.

(answer to version 3 of the question)

No. There exist bijections between $\mathbb R^n$ and $\mathbb R$, for instance.

edited May 01 '20 at 22:19

answered May 01 '20 at 22:10

Martin Argerami

205,756

I'm sorry. I mistook the concepts of injectivity and Surjectivity. – Eduardo Magalhães May 01 '20 at 22:16

score 0 · Answer 2 · answered May 01 '20 at 22:06

0

No, of course not. For example, there is a bijection

$$\mathbb{R} \to \mathbb{R}^2$$

answered May 01 '20 at 22:06

J. De Ro

21,438

$V$ and $W$ be two vector spaces such that $\dim V > \dim W$, then $T:V\longrightarrow W$ can't be Injective.

2 Answers2